Advanced Calculus Single Variable

4.1 General Considerations

The concept of a function is that of something which gives a unique output for a given
input.

Definition 4.1.1Consider two sets, D and R along with a rule which assignsa unique element of R to every element of D. This rule is called a function and it isdenoted by a letter such as f. The symbol, D

(f)

= D is called thedomain of f. Theset R, also written R

(f)

, is called therange of f. The set of all elements of R whichare of the form f

(x)

for some x ∈ D is often denoted by f

(D)

. When R = f

(D )

, thefunction f is said to be onto. It is common notation to write f : D

(f)

→ R to denotethe situation just described in this definition where f is a function defined on D havingvalues in R.

Example 4.1.2Consider the list of numbers,

{1,2,3,4,5,6,7}

≡ D. Define a functionwhich assigns an element of D to R ≡

{2,3,4,5,6,7,8}

by f

(x)

≡ x+1 for each x ∈ D.

In this example there was a clearly defined procedure which determined the function.
However, sometimes there is no discernible procedure which yields a particular
function.

Example 4.1.3Consider the ordered pairs,

(1,2)

,

(2,− 2)

,

(8,3)

,

(7,6)

and let

D ≡ {1,2,8,7},

the set of first entries in the given set of ordered pairs, R ≡

{2,− 2,3,6}

, the set of secondentries, and let f

(1)

= 2,f

(2)

= −2,f

(8)

= 3, and f

(7)

= 6.

Sometimes functions are not given in terms of a formula. For example, consider
the following function defined on the positive real numbers having the following
definition.

This is a very interesting function called the Dirichlet function. Note that it is not defined
in a simple way from a formula.

Example 4.1.5Let D consist of the set of people who have lived on the earth exceptfor Adam and for d ∈ D, let f

(d)

≡ the biological father of d. Then f is a function.

This function is not the sort of thing studied in calculus but it is a function just the same.
When D

(f )

is not specified, it is understood to consist of everything for which f makes
sense. The following definition gives several ways to make new functions from old
ones.

Definition 4.1.6Let f,g be functions with values in F. Let a,b be points of F. Thenaf + bg is the name of a function whose domain is D

(f )

∩ D

(g)

which is definedas

(af + bg)(x) = af (x) +bg (x).

The function fg is the name of a function which is defined on D

(f)

∩ D

(g)

givenby

(fg)(x) = f (x)g (x).

Similarly for k an integer, fkis the name of a function defined as

fk(x) = (f (x))k

The function f∕g is the name of a function whose domain is

D (f)∩ {x ∈ D (g) : g(x) ⁄= 0}

defined as

(f∕g)(x) = f (x)∕g (x).

If f : D

(f)

→ X and g : D

(g)

→ Y, then g ∘ f is the name of a function whose domainis

{x ∈ D (f) : f (x ) ∈ D (g)}

which is defined as

g ∘f (x ) ≡ g(f (x)).

This is called the composition of the two functions.

You should note that f

(x )

is not a function. It is the value of the function at the point x.
The name of the function is f. Nevertheless, people often write f

(x )

to denote a function
and it doesn’t cause too many problems in beginning courses. When this is done,
the variable x should be considered as a generic variable free to be anything in
D

(f)

.

Sometimes people get hung up on formulas and think that the only functions of
importance are those which are given by some simple formula. It is a mistake to think this
way. Functions involve a domain and a range and a function is determined by what it does.
This is an old idea. See Luke 6:44 where Jesus says essentially that you know a tree
by its fruit. See also Matt. 7 about how to recognize false prophets. You look at
what it does to determine what it is. As it is with people and trees, so it is with
functions.

Example 4.1.7Let f

(t)

= t and g

(t)

= 1 + t. Then fg : ℝ → ℝis given by

fg (t) = t (1 + t) = t + t2.

Example 4.1.8Let f

(t)

= 2t + 1 and g

(t)

=

√ ----
1+ t

. Then

----------
g ∘f (t) = ∘ 1+ (2t+ 1) = √2t-+-2

for t ≥−1. If t < −1 the inside of the square root sign is negative so makes no sense.Therefore, g ∘ f :

{t ∈ ℝ : t ≥ − 1}

→ ℝ.

Note that in this last example, it was necessary to fuss about the domain of g ∘f because
g is only defined for certain values of t.

The concept of a one to one function is very important. This is discussed in the following
definition.

Definition 4.1.9For any function f : D

(f)

⊆ X → Y, define the following setknown as the inverse image of y.

f−1(y) ≡ {x ∈ D (f) : f (x) = y} .

There may be many elements in this set, but when thereis always only one elementin this set for all y ∈ f

(D (f))

, the function f is one to one sometimes written,
1 − 1. Thus f is one to one, 1 − 1, if whenever f

(x)

= f

(x1)

, then x = x1. If fis one to one, the inverse function f−1is defined on f